I am doing this as exam practice so the exact answer is not as important as the proper solution. So far I have figured I could use normal distribution with the Central Limit Theorem to find the answer. I found the mean by multiplying 400 * .4 but am stuck from here, primarily finding the standard deviation.

2 Answers
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Note that subsequent runs of the experiment don't affect the probability of a success. This is definitely a binomial distribution where your experiment is flipping a coin, and your 'success' is getting a heads. You want to know what the probability of there being less than 150 successes (call this $x$) after performing the experiment 400 times (call this $n$), with the probability of a single success being 0.4 (call this $p$). To get this value you would generally use the Binomial CDF, but calculating that with such large numbers is really hard.

As you have noted, we instead need to estimate its probability with something that's easier to compute. Recall that for a large enough sample size ($n$) the normal distribution with $\mu = n \times p$ and standard deviation $ \sqrt{np(1-p)}$ approximates the binomial. So let's skew 150 on this scaled and shifted normal distribution to a value on the standard normal curve:

$$\frac{150-(400 \times 0.4)}{\sqrt{(400 \times 0.4 )(1-0.4)}}= -1.021$$
Now we need to look this value up in a Normal CDF.

Note that if $p$ had been much smaller or much larger it would have been appropriate to use a Poisson distribution with parameter $\lambda=np$.

I am not sure what you mean by involving the Central Limit Theorem, since that deals with things like the distribution of many sample means, not just one individual sample as we have in this experiment.